Estimate of the attractive velocity of attractors for some dynamical systems

Existence of polynomial attractor for a class of extensible beams with nonlocal weak damping and critical nonlinearity

“…And we also notice that it is difficult to obtain the exponential decay estimate with respect to noncompactness measure for some degenerate infinite-dimensional dynamical systems, so is it possible to reach the polynomial decay rate? Based on this observation, we put forward the more general concepts of the polynomial decay with respect to noncompactness measure and polynomial attractor in earlier studies [24,25].…”

“…In previous work [24,25], we first present an abstract theorem on estimating polynomial decay rate of noncompactness measure of bounded sets for infinite-dimensional dynamical systems and then apply the theorem to a class of wave equation with nonlocal weak damping when the growth exponent 𝜚 of the nonlinearity 𝑓 (u) is up to the subcritical case in natural energy space. However, it is failed to obtain the existence of a polynomial attractor by the method in Zhao et al [25] when 𝑓 is critical growth; the reason is that the corresponding Sobolev embedding is no longer compact. To overcome the difficulty, we will establish the existence of polynomial attractors based on contractive function [26,27], which involves some rather weak compactness associated with the repeated limit inferior and requires no compactness.…”

“…In previous work [24, 25], we first present an abstract theorem on estimating polynomial decay rate of noncompactness measure of bounded sets for infinite‐dimensional dynamical systems and then apply the theorem to a class of wave equation with nonlocal weak damping when the growth exponent $$$ \varrho $$$ of the nonlinearity $$$ f(u) $$$ is up to the subcritical case in natural energy space. However, it is failed to obtain the existence of a polynomial attractor by the method in Zhao et al [25] when $$$ f $$$ is critical growth; the reason is that the corresponding Sobolev embedding is no longer compact. To overcome the difficulty, we will establish the existence of polynomial attractors based on contractive function [26, 27], which involves some rather weak compactness associated with the repeated limit inferior and requires no compactness.…”